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Using the Properties of Trapezoids

Using the Properties of Trapezoids

A trapezoid is four-sided figure, a quadrilateral, with two sides that are parallel and two sides that are not. The parallel sides are called the bases. We call the length of the smaller base \(b,\) and the length of the bigger base \(B.\) The height, \(h,\) of a trapezoid is the distance between the two bases as shown in the figure below.

A trapezoid is shown. The top is labeled b and marked as the smaller base. The bottom is labeled B and marked as the larger base. A vertical line forms a right angle with both bases and is marked as h.

A trapezoid has a larger base, \(B,\) and a smaller base, \(b.\) The height \(h\) is the distance between the bases.

The formula for the area of a trapezoid is:

\({\text{Area}}_{\text{trapezoid}}=\frac{1}{2}h\left(b+B\right)\)

Splitting the trapezoid into two triangles may help us understand the formula. The area of the trapezoid is the sum of the areas of the two triangles. See the figure below.

An image of a trapezoid is shown. The top is labeled with a small b, the bottom with a big B. A diagonal is drawn in from the upper left corner to the bottom right corner.

Splitting a trapezoid into two triangles may help you understand the formula for its area.

The height of the trapezoid is also the height of each of the two triangles. See the figure below.

An image of a trapezoid is shown. The top is labeled with a small b, the bottom with a big B. A diagonal is drawn in from the upper left corner to the bottom right corner. There is an arrow pointing to a second trapezoid. The upper right-hand side of the trapezoid forms a blue triangle, with the height of the trapezoid drawn in as a dotted line. The lower left-hand side of the trapezoid forms a red triangle, with the height of the trapezoid drawn in as a dotted line.

The formula for the area of a trapezoid is

This image shows the formula for the area of a trapezoid and says “area of trapezoid equals one-half h times smaller base b plus larger base B).

If we distribute, we get,

The top line says area of trapezoid equals one-half times blue little b times h plus one-half times red big B times h. Below this is area of trapezoid equals A sub blue triangle plus A sub red triangle.

Definition: Properties of Trapezoids

  • A trapezoid has four sides as shown above.
  • Two of its sides are parallel and two sides are not.
  • The area, \(A,\) of a trapezoid is \(\text{A}=\frac{1}{2}h\left(b+B\right)\).

Example

Find the area of a trapezoid whose height is 6 inches and whose bases are \(14\) and \(11\) inches.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information. .
Step 2. Identify what you are looking for. the area of the trapezoid
Step 3. Name. Choose a variable to represent it. Let \(A=\text{the area}\)
Step 4.Translate.
Write the appropriate formula.
Substitute.
.
Step 5. Solve the equation. .
Step 6. Check: Is this answer reasonable?

If we draw a rectangle around the trapezoid that has the same big base \(B\) and a height \(h,\) its area should be greater than that of the trapezoid.

If we draw a rectangle inside the trapezoid that has the same little base \(b\) and a height \(h,\) its area should be smaller than that of the trapezoid.

A table is shown with 3 columns and 4 rows. The first column has an image of a trapezoid with a rectangle drawn around it in red. The larger base of the trapezoid is labeled 14 and is the same as the base of the rectangle. The height of the trapezoid is labeled 6 and is the same as the height of the rectangle. The smaller base of the trapezoid is labeled 11. Below this is A sub rectangle equals b times h. Below is A sub rectangle equals 14 times 6. Below is A sub rectangle equals 84 square inches. The second column has an image of a trapezoid. The larger base is labeled 14, the smaller base is labeled 11, and the height is labeled 6. Below this is A sub trapezoid equals one-half times h times parentheses little b plus big B. Below this is A sub trapezoid equals one-half times 6 times parentheses 11 plus 14. Below this is A sub trapezoid equals 75 square inches. The third column has an image of a trapezoid with a red rectangle drawn inside of it. The height is labeled 6. Below this is A sub rectangle equals b times h. Below is A sub rectangle equals 11 times 6. Below is A sub rectangle equals 66 square inches.

The area of the larger rectangle is \(84\) square inches and the area of the smaller rectangle is \(66\) square inches. So it makes sense that the area of the trapezoid is between \(84\) and \(66\) square inches

Step 7. Answer the question. The area of the trapezoid is \(75\) square inches.

Example

Find the area of a trapezoid whose height is \(5\) feet and whose bases are \(10.3\) and \(13.7\) feet.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information. .
Step 2. Identify what you are looking for. the area of the trapezoid
Step 3. Name. Choose a variable to represent it. Let A = the area
Step 4.Translate.
Write the appropriate formula.
Substitute.
.
Step 5. Solve the equation. .
Step 6. Check: Is this answer reasonable?
The area of the trapezoid should be less than the area of a rectangle with base 13.7 and height 5, but more than the area of a rectangle with base 10.3 and height 5.
An image of a trapezoid is shown with a red rectangle drawn around it. The larger base of the trapezoid is labeled 13.7 ft. and is the same as the base of the rectangle. The height of both the trapezoid and the rectangle is 5 ft. Next to this is an image of a trapezoid with a black rectangle drawn inside it. The smaller base of the trapezoid is labeled 10.3 ft. and is the same as the base of the rectangle. Below the images is A sub red rectangle is greater than A sub trapezoid is greater than A sub rectangle. Below this is 68.5, 60, and 51.5.
Step 7. Answer the question. The area of the trapezoid is 60 square feet.

Example

Vinny has a garden that is shaped like a trapezoid. The trapezoid has a height of \(3.4\) yards and the bases are \(8.2\) and \(5.6\) yards. How many square yards will be available to plant?

Solution

Step 1. Read the problem. Draw the figure and label it with the given information. .
Step 2. Identify what you are looking for. the area of a trapezoid
Step 3. Name. Choose a variable to represent it. Let A = the area
Step 4.Translate.
Write the appropriate formula.
Substitute.
.
Step 5. Solve the equation. .
Step 6. Check: Is this answer reasonable?
Yes. The area of the trapezoid is less than the area of a rectangle with a base of 8.2 yd and height 3.4 yd, but more than the area of a rectangle with base 5.6 yd and height 3.4 yd.
This image is a table with two rows. the first row is split into three columns. The first column is the formula Area of a rectangle equals base times height. On the next line under this it has numbers plugged into the formula; the base, 8.2 in parentheses times the height 3.4 in parentheses. Under this is it has “equals 27.88 yards squared”. The center column includes the formula of a trapezoid and says Area of a trapezoid equals one half times 3.5 yards in parentheses times 5.8 plus 8.2 in parentheses. Under this it has “equals 23.46 yards squared”. In the third column it it has the formula the area of a rectangle equals base times height. Under this it has equals 5.6 in parentheses times 3.4 in parentheses. Under this it has “equals 19.04 yards squared.” In the second row, centered from left to right it has “Area of a rectangle” and a “greater than” sign, “Area of a trapezoid” and a greater than sign and “area of a rectangle”. Under Area of a rectangle it has 27.88, then 23.46 under “area of a trapezoid”, then 19.04 under “area of a rectangle”.
Step 7. Answer the question. Vinny has 23.46 square yards in which he can plant.

Optional Video: Area of a Trapezoid

This lesson is part of:

Math Models and Geometry I

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